Domain and Range of Radical Functions

Objective

SWBAT find the domain and range of transformed radical functions.

Big Idea

Introduce these important ideas in the context of a function that has a limited domain and a limited range--students develop their own notation and conventions to show the domain and range of a function.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Warm-Up

30 minutes

Problem (1) is presented as another opportunity for students to discuss the idea that the square root function has a restricted range because we only use “half” the outputs. The graphs shown in this picture also provide a reference when thinking about domains and ranges, which is the big idea of today’s lesson.

Problem (2) is essential to prepare students for the lesson, and the language used is deliberately un-intimidating. Ask students to justify each of their answers:

How do you know this input is not allowed?

How do you know an output is impossible?

The quadratic function obviously has no “impossible inputs,” but it is presented this way so that students have to convince themselves that all inputs are allowed. Ask them:

Why can we put any number into a quadratic function?

Why can’t we get any number out of a radical function?

The words “domain” and “range” are introduced in problem (3) and once students have a sense of what these words mean, I ask them:

How could you show all possible inputs?

How could you show all possible outputs?

Students will be provided with number lines to show domains and ranges during the lesson, which is one efficient way. They may also use inequalities, or just verbal descriptions for now. I ask them:

What would be the best way to show the domain and range?

These three problems effectively give students the chance to make sense of most of the day’s lesson on their own. If you find some students floundering during this time, I reassign them a new partner or ask students who have some ideas to convince their peers. I have found that when a struggling student is put in the position of trying to decide whether or not they agree with a student who arrived at an answer more quickly, this helps change the “status” in the classroom, because now the struggling student has an important job. Again, this is only possible if you give no indication at all of whether or not you agree with a student’s answers. I say, “____________, can you convince ______________ that your answers to problem (2) make sense?” This brings in both MP1 and MP2.

Domain and Range Warm-Up.docx

Investigation and New Learning

30 minutes

Domain and Range intro questions.docx

Big Ideas Questions.docx

Domain and Range of Radical Functions Problem Set.pdf

Domain and Range of Radical Functions Problem Set.docx

Domain and Range Generalization Exceeds.docx

Closing

10 minutes

The closing gives students another opportunity to articulate the same big ideas that we have been thinking about for a while. One interesting idea to highlight is related to questions (A) and (B).

You would think that the domain of the original function becomes the range of the inverse function, but this is not the case when it comes to quadratic and radical functions. It is true that the range of the quadratic function becomes the domain of its inverse, however. This is a deeper conversation to have with students who have already developed a deeper understanding.

Questions (C) and (D) are important to ensure that students don’t just start memorizing things. We want them not only to be able to find domains and ranges of radical functions concretely, but also to explain more abstractly the ideas behind what they are doing.

The restrictions on the domain come from the fact that there is no square root of a negative number. I like to push students further: Why can’t we? They should be able to say that the “square root” of a number is the number that we can square to get the original number, but any number that we square will result in a positive output. The restrictions on the range come from the fact that we need to restrict the outputs in order for the radical function to be a function. In other words, only one output is allowed for each input.

These conversations are what elevate the course to a higher level of thinking and help students develop deep and abstract understandings. Even if it is difficult to ensure that all students understand these big ideas each day, the more frequently that you have these conversations, the more this way of thinking will become part of the culture of the class and students will start to develop the idea that “doing math” does not mean simply working with numbers to get answers, but actually to make sense of what is going on and to create deeper generalizations.